276 
STATISTICS: PEARL AND REED 
Proc. N. a. S. 
inaccurate for the United States, at least, for any considerable period of 
time. What actually happens is that following any census estimates are 
made by one or another of these methods of the population for each year 
up to the next census, on the basis of data given by the last two censuses 
only. When that next census has been made, the previous estimates 
of the inter-censal years are corrected and adjusted on the basis of the 
facts brought out at that census period. 
Obviously the best general method of estimating population in inter- 
censal years is that of fitting an appropriate curve to all the available 
data, and extrapolating for years beyond the last census, and reading off 
from the curve values for inter-censal years falling between earlier 
censuses. The methods of arithmetic or geometric progression use only 
two census counts at the most. Fitting a curve to all the known data 
regarding population by the method of least squares must obviously 
give a much sounder and more accurate result. In making this state- 
ment, one realizes perfectly, of course, the dangers of extrapolation. 
These dangers have been well emphasized by Perrin,^ who used higher 
order parabolas to predict the future population of Buenos Aires. In 
keeping sharply before our minds the dangers of extrapolation from a 
curve, we are apt to forget that the methods of extrapolation by arithmetic 
or geometric progression have much less general validity than from a 
curve, and the inaccuracies are found in practice, except by the rarest 
of accidents, to be actually greater. 
The first one to attempt an adequate mathematical representation of 
the normal rate of growth of the population of the United States was 
Pritchett.^ Taking the census data from 1790 to 1880, inclusive, Pritchett 
fitted by the method of least squares the following equation: 
P = A + 5^ + 02 + Dt^ (i) 
where P represents the population and t the time from some assumed 
epoch. As a matter of fact, Pritchett took the origin of the curve at 
1840, practically the center of the series. With this third-order parabola 
Pritchett got a very accurate representation of the population between 
the dates covered. As will presently appear this curve did not ^ give, 
even within the period covered, as accurate results as a more adequate 
curve would have done, and it overestimated the population after a very 
short interval beyond the last observed ordinate as is shown in table 2. 
Some 13 years ago one of the writers^ demonstrated the applicability 
of a logarithmic curve of the form 
y = a -\- h% cx"^ d log x (ii) 
to the representation of growth changes, using the aquatic plant Cera- 
tophyllum as material. Following the application of this curve to growth 
of this plant it was found equally useful in representing a wide range of 
other growth and related changes.^ This list now includes, of matters 
